Evaluating thermal expectation values by almost ideal sampling with Trotter gates

We investigate the sampling efficiency for the simulations of quantum many-body systems at finite temperatures when initial sampling states are generated by applying Trotter gates to random phase product states (RPPSs). We restrict the number of applications of Trotter gates to be proportional to the system size, and thus the preparation would be easily accomplished in fault-tolerant quantum computers. When the Trotter gates are made from a nonintegrable Hamiltonian, we observe that the sampling efficiency increases with system size. This trend means that almost ideal sampling of initial states can be achieved in sufficiently large systems. We also find that the sampling efficiency is almost equal to that obtained by a thermal pure quantum (TPQ) state method utilizing Haar random sampling in some cases. These findings suggest that chaotic Hamiltonian dynamics can transform RPPSs into an alternative to TPQ states for evaluating thermal expectation values.

Figure 1

System-size dependence of the average entanglement entropy of sampled initial states (top panel) and the efficiency (bottom panel) obtained from 1024 samples for the TPQ states (blue solid line), the RPPSs evolved by the Trotter gates obtained from the XXZ chain without the staggered magnetic field (orange dashed line), and those evolved by the Trotter gates obtained from the XXZ chain with the staggered magnetic field (green dot-dashed line). The inverse temperature $\beta$ is set to $3.0J^{-1}$. The error bars indicate $1\sigma$ uncertainty. The $1\sigma$ uncertainty of the efficiency is estimated from the bootstrap analysis with 4000 resampled data.

Figure 2

System-size and inverse temperature dependence of the efficiency obtained from 1024 samples for the RPPSs evolved by the Trotter gates obtained from the XXZ chain with (a) and without (b) the staggered magnetic field.

Figure 3

System-size dependence of the difference of the efficiency from unity. The efficiencies are the same with those shown in the bottom panel of Fig. 1.

Figure 4

System-size dependence of the difference of the efficiency from unity for some temperatures. The efficiencies are obtained from 1024 samples for the RPPSs evolved by the Trotter gates obtained from the XXZ chain with the staggered magnetic field $h=J$. The error bars indicate $1\sigma$ uncertainty estimated from the bootstrap analysis with 4000 resampled data.

Figure 5

System-size dependence of the variance of neighboring spin correlation at the center of the system. The variance is estimated from the error propagation formula (11). The settings of simulations are the same with those of Fig. 1.

Figure 6

System-size dependence of the average entanglement entropy of sampled initial states (top panel) and the efficiency (bottom panel) obtained from 1024 samples for the TPQ states (blue solid line), RPPSs evolved by the Trotter gates obtained from the transverse-field Ising chain (orange dashed line), and those evolved by the Trotter gates obtained from the mixed-field Ising chain (green dot-dashed line). The inverse temperature $\beta$ is set to $3.0J^{-1}$. The error bars indicate $1\sigma$ uncertainty. The $1\sigma$ uncertainty of the efficiency is estimated from the bootstrap analysis with 4000 resampled data.

Figure 7

Inverse-temperature dependence of the internal energies of the Heisenberg chain estimated from Eqs. (5) and (8). The top (middle) panel corresponds to the simulation with the Trotter gates obtained from the transverse-field (mixed-field) Ising chain with $h_x=J$ ($h_x=h_z=J$). The system size $L$ is set to 20. Error bars indicate $1\sigma$ uncertainty, and they are on the order of ${10}^{-4}$ and thus smaller than the line width. The differences of the internal energies per site estimated from the two equations are given in the bottom panel.

Figure 8

System-size dependence of the average entanglement entropy of sampled initial states (top panel) and the efficiency (bottom panel) obtained from 1024 samples for the TPQ states (blue solid line), RPPSs evolved by the Trotter gates obtained from the transverse-field Ising chain (orange dashed line), and those evolved by the Trotter gates obtained from the mixed-field Ising chain (green dot-dashed line). The system Hamiltonian is the mixed-field Ising chain in this simulation. Inverse temperature $\beta$ is set to $3.0J^{-1}$. The error bars indicate $1\sigma$ uncertainty. The $1\sigma$ uncertainty of the efficiency is estimated from the bootstrap analysis with 4000 resampled data.

Figure 9

Differences of the internal energies per site estimated from Eqs. (5) and (8) with different system sizes $L=10$ and 20. The system Hamiltonian and the Trotter Hamiltonian are set to the Heisenberg chain and the mixed-field Ising chain with $(h_x,h_z)=(J,J)$, respectively.